305 research outputs found
On Binary de Bruijn Sequences from LFSRs with Arbitrary Characteristic Polynomials
We propose a construction of de Bruijn sequences by the cycle joining method
from linear feedback shift registers (LFSRs) with arbitrary characteristic
polynomial . We study in detail the cycle structure of the set
that contains all sequences produced by a specific LFSR on
distinct inputs and provide a fast way to find a state of each cycle. This
leads to an efficient algorithm to find all conjugate pairs between any two
cycles, yielding the adjacency graph. The approach is practical to generate a
large class of de Bruijn sequences up to order . Many previously
proposed constructions of de Bruijn sequences are shown to be special cases of
our construction
Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields
A maximal minor of the Laplacian of an -vertex Eulerian digraph
gives rise to a finite group
known as the sandpile (or critical) group of . We determine
of the generalized de Bruijn graphs with
vertices and arcs for and , and closely related generalized Kautz graphs, extending and
completing earlier results for the classical de Bruijn and Kautz graphs.
Moreover, for a prime and an -cycle permutation matrix
we show that is isomorphic to the
quotient by of the centralizer of in
. This offers an explanation for the coincidence of
numerical data in sequences A027362 and A003473 of the OEIS, and allows one to
speculate upon a possibility to construct normal bases in the finite field
from spanning trees in .Comment: I+24 page
Characterizations of generators for modified de Bruijn sequences
AbstractOrder n modified de Bruijn sequences are created by removing a single zero from the longest run of zeros in period 2n de Bruijn sequences. The M sequences are then the natural linear subset of modified de Bruijn sequences. Recursions which are the nonlinear duals to primitive polynomials over GF(2) are developed. Data is presented for 4 β€ n β€ 6
Grein. A New Non-Linear Cryptoprimitive
In this thesis, we will study a new stream cipher, Grein, and a new cryptoprimitive used in this cipher. The second chapter gives a brief introduction to cryptography in general. The third chapter looks at stream ciphers in general, and explains the advantages and disadvantages of stream ciphers compared to block ciphers. In the fourth chapter the most important building blocks used in stream ciphers are explained. The reader is excepted to know elementary abstract algebra, as much of the results in this chapter depend on it. In the fifth chapter, the stream cipher Grain is introduced. In chapter six, the new stream cipher, Grein, is introduced. Here, we look at the different components used in the cipher, and how they operate together. In chapter seven, we introduce an alteration to the Grein cryptosystem, which hopefully have some advantagesMaster i InformatikkMAMN-INFINF39
Combinatorial and Additive Number Theory Problem Sessions: '09--'19
These notes are a summary of the problem session discussions at various CANT
(Combinatorial and Additive Number Theory Conferences). Currently they include
all years from 2009 through 2019 (inclusive); the goal is to supplement this
file each year. These additions will include the problem session notes from
that year, and occasionally discussions on progress on previous problems. If
you are interested in pursuing any of these problems and want additional
information as to progress, please email the author. See
http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019,
fixed a few issues from some presenters 6/29/201
Efficient indexing of necklaces and irreducible polynomials over finite fields
We study the problem of indexing irreducible polynomials over finite fields,
and give the first efficient algorithm for this problem. Specifically, we show
the existence of poly(n, log q)-size circuits that compute a bijection between
{1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials
of degree n over a finite field F_q. This has applications in pseudorandomness,
and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP].
Our approach uses a connection between irreducible polynomials and necklaces
( equivalence classes of strings under cyclic rotation). Along the way, we give
the first efficient algorithm for indexing necklaces of a given length over a
given alphabet, which may be of independent interest
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